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Continuity of a multivariable function

  1. Oct 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Given the function:

    x*y / (4-x²-2y²) if x²+2y² ≠4
    0 if x²+2y² = 4

    Check if the function is continuous.



    2. Relevant equations



    3. The attempt at a solution

    I tried using various ways to see if the result of the limit as (x,y)→(2,0) was the same, such as y=x-2, y=(x-2)², etc..
    I didn't manage to prove that the limit didn't existed. I always arrive at the 0/0 indetermination...

    If anyone could point me in the right direction I'd appreciate!

    D.
     
  2. jcsd
  3. Oct 23, 2012 #2
    convert to ellyptic coordinates x=2rcos(t); y=sqt(2)rsin(t) and compute the limit as r tends to 0,using L'hopital's rule.If the limit is not identically 0 (or depends on t),then the function is not continuous at the poins on the ellipse.
     
  4. Oct 23, 2012 #3
    Hmm I'll try doing it. However I never learned ellyptic coordinates. I'm wondering if there's any other way to solve this!
     
    Last edited: Oct 23, 2012
  5. Oct 23, 2012 #4
    o.k don't name it,just substitute in the function and compute the limit as r tends to 1 and t is constant.
     
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